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# A Further Study on Using x^{·} = λ[αR + βP] (P = F − R(F·R) / ||R||^{2}) and x^{·} = λ[αF + βP^{∗}] (P^{∗} = R − F(F·R) / ||F||^{2}) in Iteratively Solving the Nonlinear System of Algebraic Equations F(x) = 0

Center for Aerospace Research & Education, University of California, Irvine

Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw

*Computer Modeling in Engineering & Sciences* **2011**, *81*(2), 195-228. https://doi.org/10.3970/cmes.2011.081.195

## Abstract

In this continuation of a series of our earlier papers, we define a hyper-surface*h*(x,

*t*) = 0 in terms of the unknown vector x, and a monotonically increasing function

*Q(t)*of a time-like variable t, to solve a system of nonlinear algebraic equations

**F(x)**=

**0**. If

**R**is a vector related to ∂h / ∂x, , we consider the evolution equation

**x**, where

^{·}= λ[αR + βP]**P = F − R(F·R) / ||R||**such that

^{2}**P·R**= 0; or

**x**, where

^{·}= λ[αF + βP^{∗}]**P**such that

^{∗}= R − F(F·R) / ||F||^{2}**P**= 0. From these evolution equations, we derive Optimal Iterative Algorithms (OIAs) with Optimal Descent Vectors (ODVs), abbreviated as ODV(R) and ODV(F), by deriving optimal values of

^{*}·F*α*and

*β*for fastest convergence. Several numerical examples illustrate that the present algorithms converge very fast. We also provide a solution of the nonlinear Duffing oscillator, by using a harmonic balance method and a post-conditioner, when very high-order harmonics are considered.

## Keywords

## Cite This Article

Liu, C., Dai, H., Atluri, S. N. (2011). A Further Study on Using x^{·}= λ[αR + βP] (P = F − R(F·R) / ||R||

^{2}) and x

^{·}= λ[αF + βP

^{∗}] (P

^{∗}= R − F(F·R) / ||F||

^{2}) in Iteratively Solving the Nonlinear System of Algebraic Equations F(x) = 0.

*CMES-Computer Modeling in Engineering & Sciences, 81(2)*, 195–228.